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The number of solutions of a random system of polynomials over a finite field
Ritik Jain · 2024-09-11 · via math.PR updates on arXiv.org

We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$. Then, in the case that $R$ is a field, under a necessary-and-sufficient condition on the sample space, we show that the number of common zeros is binomially distributed.

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